Appendix 2: Electromagnetic Plane Waves
185
It remains to determine the polarization vector ν , which must be consistent with
the Lorentz condition (11.82). With the solution in (11.86) this condition is
β k β = 0.
(11.87)
If we align the z axis along the space part of the null vector we may take U = ct − z
or some constant multiple as in Sect. 11.3. We may choose the polarization to
be along either the x or y direction and thus obtain the solution in terms of either
components or unit vectors as
A ν = (0, A 1 (U ), A 2 (U ), 0) = A 1 (U )ˆ e 1 + A 2 (U )ˆ e 2 .
(11.88)
The picture we thereby obtain is that the polarization vector has no time component
and points along either the x or y direction, perpendicular to the propagation direction
z of the wave. The wave is therefore called transverse.
A transformation to what we will call a tilde gauge is defined in terms of a scalar
function ϕ as
A ν = A ν + ϕ ,ν .
(11.89)
This does not change the antisymmetric Maxwell electromagnetic field tensor, which
is related to the vector potential by
F μν = A μ,ν − A ν,μ .
(11.90)
Moreover, such a gauge change can take us between various choices of the
polarization.
By using a gauge transformation we can put any solution of the equations (11.85)
into the transverse form (11.88). In terms of U = k β x
β the solution and Lorentz
gauge condition are
A
μ
= A
μ
(U ), A
ν ,ν =
∂ A
ν
∂ x ν =
∂U
∂ x ν
∂ A
ν
∂U
= A
ν ,U k ν = 0.
(11.91)
Here the notation ,U denotes differentiation with respect to the argument U . For
the gauge function ϕ we choose some function of U to be determined; because of
(11.89) the function ϕ must be a solution of the wave equation,
2
ϕ = ϕ ,λ
,λ
= 0.
(11.92)
In the tilde system the vector potential and its divergence are then